cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A119950 Row sums of (denominator) triangle A119948.

Original entry on oeis.org

1, 8, 45, 160, 1025, 936, 7693, 22016, 123201, 181000, 2190221, 2409408, 33932665, 43909880, 29887229, 68917248, 1244818081, 732937032, 14699459725, 17464448672, 14290498033, 26212519432, 630291945053
Offset: 1

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Formula

a(n)=sum(A119948(n,m),m=1..n), n>=1.

A027446 Triangle read by rows: square of the lower triangular mean matrix.

Original entry on oeis.org

1, 3, 1, 11, 5, 2, 25, 13, 7, 3, 137, 77, 47, 27, 12, 147, 87, 57, 37, 22, 10, 1089, 669, 459, 319, 214, 130, 60, 2283, 1443, 1023, 743, 533, 365, 225, 105, 7129, 4609, 3349, 2509, 1879, 1375, 955, 595, 280, 7381, 4861, 3601, 2761, 2131, 1627, 1207, 847, 532, 252
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Numerators of nonzero elements of A^2, written as rows using the least common denominator, where A[i,j] = 1/i if j <= i, 0 if j > i. [Edited by M. F. Hasler, Nov 05 2019]

Examples

			Triangle starts
     1
     3,    1
    11,    5,    2
    25,   13,    7,    3
   137,   77,   47,   27,   12
   147,   87,   57,   37,   22,   10
  1089,  669,  459,  319,  214,  130,  60
  2283, 1443, 1023,  743,  533,  365, 225, 105
  7129, 4609, 3349, 2509, 1879, 1375, 955, 595, 280
  ... - _Joerg Arndt_, Mar 29 2013

Links

Crossrefs

The row sums give A081528(n), n>=1.
The column sequences give A025529, A027457, A027458 for j=1..3.
The diagonal sequences give A002944, A027449, A027450.
Cf. A027447, A027448.

Programs

  • Mathematica
    rows = 10;
M = MatrixPower[Table[If[j <= i, 1/i, 0], {i, 1, rows}, {j, 1, rows}], 2];
T = Table[M[[n]]*LCM @@ Denominator[M[[n]]], {n, 1, rows}];
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 05 2013, updated May 06 2022 *)
  • PARI
    A027446_upto(n)={my(M=matrix(n, n, i, j, (j<=i)/i)^2); vector(n,r,M[r,1..r]*denominator(M[r,1..r]))} \\ M. F. Hasler, Nov 05 2019

Formula

The rational matrix A^2, where the matrix A has elements a[i,j] = 1/A002024(i,j), is equal to A119947(i,j)/A119948(i,j).
a(i,j) = lcm(seq(A119948(i,m),m=1..i))*A119947(i,j)/A119948(i,j), 1 <= j =< i and zero otherwise.

Extensions

Edited by M. F. Hasler, Nov 05 2019

A119947 Triangle of numerators in the square of the matrix A[i,j] = 1/i for j <= i, 0 otherwise.

Original entry on oeis.org

1, 3, 1, 11, 5, 1, 25, 13, 7, 1, 137, 77, 47, 9, 1, 49, 29, 19, 37, 11, 1, 363, 223, 153, 319, 107, 13, 1, 761, 481, 341, 743, 533, 73, 15, 1, 7129, 4609, 3349, 2509, 1879, 275, 191, 17, 1, 7381, 4861, 3601, 2761, 2131, 1627, 1207, 121, 19, 1, 83711, 55991, 42131, 32891, 25961
Offset: 1

Views

Author

Wolfdieter Lang, Jul 20 2006

Keywords

Comments

The triangle of the corresponding denominators is A119948. The rationals appear in lowest terms (while in A027446 they are row-wise on the least common denominator).
The triangle with row number i multiplied with the least common multiple (LCM) of its denominators yields A027446.
First column is A001008. - Tilman Neumann, Oct 01 2008
Column 2 is A064169. - Clark Kimberling, Aug 13 2012
Third diagonal (11, 13, 47, ...) is A188386. - Clark Kimberling, Aug 13 2012

Examples

			The rationals are [1]; [3/4, 1/4]; [11/18, 5/18, 1/9]; [25/48, 13/48, 7/48, 1/16]; ... See the W. Lang link for more.
From _Clark Kimberling_, Aug 13 2012: (Start)
As a triangle given by f(n,m) = Sum_{h=m..n} 1/h, the first six rows are:
    1
    3    1
   11    5    1
   25   13    7    1
  137   77   47    9    1
   49   29   19   37   11    1
  363  223  153  319  107   13    1
(End)

Links

Crossrefs

Cf. A002024: i appears i times (denominators in row i of the matrix A).
Row sums give A119949. Row sums of the triangle of rationals always give 1.
For the cube of this matrix see the rational triangle A119935/A119932 and A027447; see A027448 for the fourth power.
Cf. A001008, A027446, A064169, A119948, A188386.

Programs

  • Mathematica
    f[n_, m_] := Numerator[Sum[1/k, {k, m, n}]]
Flatten[Table[f[n, m], {n, 1, 10}, {m, 1, n}]]
TableForm[Table[f[n, m], {n, 1, 10}, {m, 1, n}]] (* Clark Kimberling, Aug 13 2012 *)
  • PARI
    A119947_upto(n)={my(M=matrix(n,n,i,j,(j<=i)/i)^2);vector(n,r,apply(numerator,M[r,1..r]))} \\ M. F. Hasler, Nov 05 2019

Formula

a(i,j) = numerator(r(i,j)) with r(i,j):=(A^2)[i,j], where the matrix A has elements a[i,j] = 1/i if j<=i, 0 if j>i, (lower triangular).

Extensions

Edited by M. F. Hasler, Nov 05 2019

A247884 Number of positive integers < 10^n divisible by their first digit.

Original entry on oeis.org

9, 41, 327, 3158, 31450, 314349, 3143320, 31433005, 314329833, 3143298089, 31432980631, 314329806030, 3143298060001, 31432980599686, 314329805996514, 3143298059964770, 31432980599647312, 314329805996472711, 3143298059964726682, 31432980599647266367
Offset: 1

Views

Author

Derek Orr, Sep 25 2014

Keywords

Comments

a(n)/10^n seems to converge to a number around .3143...
a(n)/10^n converges to 7129/22680. - Hiroaki Yamanouchi, Sep 26 2014

Links

Crossrefs

Cf. A034837, A028569, A119947, A119948.

Programs

  • PARI
    a(n)=c=0;for(k=1,10^n-1,d=digits(k);if(k%d[1]==0,c++));c
n=1;while(n<10,print1(a(n),", ");n++)
  • Python
    count = 9 # Start with the first 9 digits
print(1, 9)
n = 2
while n < 101:
    for a in range(1, 10):
        count += 10**(n-1)//a
        if 10**(n-1) % a != 0:
            count += 1
    print(n, count)
    n += 1
# David Consiglio, Jr., Sep 26 2014

Formula

G.f.: x*(9 - 67*x + 24*x^2 + 14*x^3 - 56*x^4 + 21*x^5 + 7*x^6 + 5*x^7)/((1 - x)^2*(1 + x)*(1 - 10*x)*(1 - x + x^2)). - Robert Israel, Mar 10 2025

Extensions

a(9)-a(20) from Hiroaki Yamanouchi, Sep 26 2014
Showing 1-4 of 4 results.

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